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The prime spectrum of algebras of quadratic growth

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Original languageEnglish
Pages (from-to)414-431
Number of pages18
JournalJournal of Algebra
Volume319
Issue number1
DOIs
Publication statusPublished - 1 Jan 2008

Abstract

We study prime algebras of quadratic growth. Our first result is that if A is a prime monomial algebra of quadratic growth then A has finitely many prime ideals P such that A/P has GK dimension one. This shows that prime monomial algebras of quadratic growth have bounded matrix images. We next show that a prime graded algebra of quadratic growth has the property that the intersection of the non-zero prime ideals P such that A I P has GK dimension 2 is non-zero, provided there is at least one such ideal. From this we conclude that a prime monomial algebra of quadratic growth is either primitive or has non-zero locally nilpotent Jacobson radical. Finally, we show that there exists a prime monomial algebra A of GK dimension two with unbounded matrix images and thus the quadratic growth hypothesis is necessary to conclude that there are only finitely many prime ideals such that A/P has GK dimension 1. (C) 2007 Elsevier Inc. All rights reserved.

    Research areas

  • GK dimension, quadratic growth, primitive rings, PI rings, graded algebra, GELFAND-KIRILLOV DIMENSION, AFFINE ALGEBRAS, EXAMPLES

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